Zeal Test Series 2019: Set Theory & Algebra - Relations

Question

Comments

7 right ??

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9 ??

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$S=\left \{1,2,3,4 \right \}$

$R=\left \{ (1,2),(1,4),(2,3)\right \}$

If $R$ become Equivalence relation,then it should be$:(1)$ Reflexive relation $(2)$ Symmetric relation  $(3)$ Transitive relation  

So,for Reflexive relation should be added $(1,1),(2,2),(3,3),(4,4)$ diagonal elements  $R=\left \{ (1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,3)\right \}$

for Symmetric relation should be added $(2,1),(4,1),(3,2)$ elements  $R=\left \{ (1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,3),(2,1),(4,1),(3,2)\right \}$

for Transitive  relation should be added $(1,3),(3,1),(2,4)$ elements

  $R=\left \{ (1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,3),(2,1),(4,1),(3,2),(1,3),(3,1),(2,4)\right \}$

So i think it will be $10$

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@akshman as (4,1) and (1,2) so we need to add (4,2) also

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Bro it's 13

thisis the problem of this question that we have to check till last for each

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@lakshman (3,4),(4,2)(4,3) would also be there

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this relation will contain all the 16 order pair so 13 will be answer

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sorry i miss some Transitive relation and Symmetric relation

$R=\left \{ (1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,3),(2,1),(4,1),(3,2),(1,3),(3,1),(2,4),(4,2)(3,4),(4,3)\right \}$

So,answer is $13$

Answer 1

An easy way is by making a graph. We know an equivalence relation partitions the set into classes. Thus if u make a graph for this relation, it consists of only one partition P[1] = {1,2,3,4} = S1.

The no of ordered pairs contributed by a partiton is |S|^2. So total cardinality of R = |S1|^2  will be 4^2 = 16.

Given ordered pairs = 3

=> To be added = 16-3 = 13

Answer 2

In question it is clearly mention that it is equivalence relation and it is also given that (1,2) ,(1,4),(2,3) are pairs hence by concepts of equivalence classes you can directly see that equivalence classes of this relation will be [1,2,3,4] hence no. of ordered pairs=16 so, no. of pairs to be added is 16-3=13